We generalize the model theory of small profinite structures developed by Newelski to the case of compact metric spaces considered together with compact groups of homeomorphisms and satisfying the existence of m-independent extensions (we call them compact e-structures). We analyze the relationships between smallness and different versions of the assumption of the existence of m-independent extensions and we obtain some topological consequences of these assumptions. Using them, we adopt Newelski's proofs of various results about small profinite structures to compact e-structures. In particular, we notice that a variant of the group configuration theorem holds in this context.
A general construction of compact structures is described. Using it, a class of examples of compact e-structures which are not small is constructed.
It is also noticed that in an m-stable compact e-structure every orbit is equidominant with a product of m-regular orbits.
